Minimizing the Length of a Crease

Date: 11/11/1999 at 13:54:30
From: Jessica J. Johnston
Subject: Minimize the value of L
This is the exact problem in the book:
A rectangular sheet of 8.5 x 11-in. paper is placed on a flat surface,
and one of the corners is lifted up and placed on the opposite longer
edge. With all four corners now held fixed, the paper is smoothed
flat.
a) Make the length of the crease as small as possible (call the length
of the crease L)
b) Show that L^2 = 2x^3/(2x-8.5)
c) Minimize L^2
d) Minimize the value of L
I'm stuck on (b) because I can't figure out how to show that it is
true. I can't do the rest of them then either.

Date: 11/12/1999 at 17:06:00
From: Doctor Peterson
Subject: Re: Minimize the value of L
Hi, Jessica.
I like this problem, but because you didn't say what x was, I had to
play with it for a day before I solved it. Every choice I made for x
resulted in complex equations, until I chose the right one, for which
your equation "fell out" of the drawing without too much work. But
it's still not easy; you have to work through three different
triangles and use Pythagoras twice to get the formula.
Here's my picture:
D E A
+-------+--------------+
| W-x / \ x |
| / \ |
| / \ |
y| /x \ |
| / \ |
| / \ |
|/ L\ |y+z
F+ \ |
| \ \ |
| \ \ |
z| \ \ |
| y+z \ \ |
| \ \ |
| \ \|
C+----------------------+B
| W |
| |
We want to find L in terms of x. F is the position of corner A after
the fold, so EFB is a right triangle congruent to EAB. I've added a
line BC parallel to AD, forming a rectangle, and called its length
(the width of the paper, 8.5 inches) W. Then I labeled DE as W-x, and
introduced variables y and z, whose sum is the length of AB, which
you'll find on the way to the answer.
There are two important pairs of triangles. First, EAB and EFB are
congruent, as I said, because they are the same part of the paper
before and after folding. (You can make that more geometrical if you
want.) That lets me label EF and BF as x and y+z. Second, EDF and FCB
are similar; look at their angles.
You should first find y in terms of x using Pythagoras in triangle
EDF. Then you can use the similar triangles to get z in terms of x.
Finally, you can use Pythagoras to get L^2 in terms of x, y, and z,
and substitute to get it in terms of x alone. There's some algebra
involved, but everything simplifies neatly to give the formula you
want:
L^2 = 2x^3/(2x-W)
When you actually solve the problem, you'll want to check whether
the solution is a crease that crosses the bottom of the paper, with
y+z > 11. Unless that happens, the formula you get will be sufficient,
but if it does, the crease will be shorter than the L you calculate.
- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/